Can the opponent's turn affect the reward for a DQN agent action?
Does the opponent's turn affect the calculated rewards? 


Yes, in general it can. Obvious case, in a two player game where the opponent could win or lose on their turn, but has other options.


  As far as I know, the reward should only be the result of the agent's action right?


In a well-defined MDP, the reward should be a stochastic function of the current state and the agent's action. The stochastic part can include any changes due to an opponent player.

If the opponent player is random, or follows a well-defined and fixed policy, then you consider them part of the environment. So this requirement is met technically. The reward does only depend on current state and the agent's action. The actual result may happen on the opponent's turn, but that does not matter. 

In a card game where the opponent's cards are hidden and affect their strategy, this may not strictly be the case, because the visible state will not determine the opponent's behaviour. The problem stops being an MDP, and starts being a POMDP. Whether or not that impacts the agent will depend on how much strategy relies on the hidden nature of these cards. In blackjack, there is little impact to not knowing an opponent's cards before they are played out - there is little difference between hidden cards and cards that are randomly determined after the agent plays. So you can get away with pretending it is a normal MDP. In poker, the knowledge of hidden cards is almost everything about the game, so a POMDP or other approach that tracks possible hidden state is required.

Note that learning to defeat a random or expert player is usually not the same as learning to play optimally (unless your expert player is already optimal). For that you may need self-play and an agent which learns both players' policies.